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We consider the autoregressive model on $\R^d$ defined by the followingstochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d.random variables valued in $\R^d\times \R^+$. The critical case, when$\E\big[\log A_1\big]=0$, was studied by Babillot, Bougeorol and Elie, whoproved that there exists a unique invariant Radon measure $\nu$ for the Markovchain $\{X_n \}$. In the present paper we prove that the weak limit of properlydilated measure $\nu$ exists and defines a homogeneous measure on$\R^d\setminus \{0\}$.

On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case